DSLRs are good for things like that, with their large sensor and fast lenses you can throw that background right out! Once again, as Michael mentioned, the key is to use the lowest F number. The F number is the aperture of the lens, a smaller F number (Ex: F2.0 or F2. opens up the aperture of the lens wider, so you can use a shorter exposure and it blurs the background. A higher F number (Ex: F16 or F22) makes the lens aperture smaller, forcing you to use a longer exposure, but it gets more into focus.

As you can see in this picture of my puppy here:

The background is blurred and out of focus. I took this with my Canon Powershot, which cannot blur background. So, how did I blur this one? I used the Gaussian Blur effect in Photoshop elements, a very handy tool.